Mathematics Grade 12 P2 Free State Mock 2023 Paper & Memo

Downloaded from Stanmorephysics.com Copyright reserved education Department of Education FREE STATE PROVINCE This question paper consists of 13 pages, 1 information shect and an answer book of 19 pages. Please turn over

mattawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. 10. This questioaqaper consists of 10 questions. Answer ALL‘the questions in the SPECIAL ANSWER BOOK provided. Clearly show-ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. Unless stated otherwise, you may use an approved scientific calculator (non-programmable and non-graphical). If necessary, round off answers to TWO decimal places unless stated otherwise. Diagrams are NOT necessarily drawn to scale. Number the answers correctly according to the numbering system used in this question paper. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

mattPawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. QUESTION 1 To celebrate Pi Day at school, learners participate in a competition to write down the value of Pi (7), up to the most correct decimal places. Eleven learners make it to the final round of the competiti ahere their number of correct decimal places is counted. enn The judges sop dsng after the first mistake. The results of the eleven learners are shown in the table betel 63 79 50 74 715 66 150 86 72 74 60 1.1 Calculate the: 1.1.1 Mean of the data (2) 1.1.2. The standard deviation for the given data dd) 1.1.3. Number of results that lie outside ONE standard deviation of the mean (3) 12 Identify the outlier in the given results. qd) 13 The result with fithe most correct decimal places is increased by k %, while the i er of the lowest correct decimal places is decreased by 7% emain unchanged. Only one of the opti correctly reflects the new range of the data in terms of arid Choos answer and write only the letter next to the question number in the ANSWER BOO A. 100+ k -t B. 150k - 50¢ Cc. 150k + 50t Dz 100+ 3K44; (2) 2 2 —, 1.4 It was established that a judge made a mistake with one of the six: owest results. The result was corrected and changed to double its original val “How will this change impact the median of the data? Motivate your answer. | (2) (11) Copyright reserved Please turn over

mattDewatoaded from Stanmorephysics.com Grade 12 Prep. Exams. QUESTION 2 FS/September 2023 On the first Saturday of a month, for a period of ten months, information was recorded about the temperature at midday (in °C); and the number of ice creams sold at an ice cream stand at a particular beach. The data is shown in the table below and represented on the scatter plot. This data'sléast squares regression line is drawn on the scatter plot. ‘Temperature-at midday (in °C) 16 20 23 29 33 38 40 38 | 35 30 Number of ice creams sold 12 17 19 44 64 70 74 66 | 60 40 SCATTER PLOT 75 70 65 60 55 50 45 40 35 Number of ice creams sold 30 25 20 15 10 15 20 25 30 Temperature at midday (in °C) 35 40 Copyright reserved Please turn over

mattawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. 2.1 Refer to the scatter plot. Would you say that the relationship between the temperature at midday and the number of ice creams sold is weak or strong? Motivate your answer. (2) 2.2 Determitieithe equation of the least squares regression line. (3) 2.3 Predict{h@/number of ice creams that will be sold on a Saturday if the temperature is 26 °C at midday. (2) 2.4 On another first Saturday of the month, the temperature at midday was 24 °C and 40 ice creams were sold. If this data is added to the data set, how will the prediction of the number of ice creams sold within the given domain be affected? Motivate your answer. (2) [9] Copyright reserved Please turn over

mattPawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. QUESTION 3 In the diagram below, A(2 ; 4), O, B(6 ; 2) and C are the vertices of a quadrilateral. D and E are thesmidpoints of AC and BC, respectively. F(6 3 4 is a point on DE. From B the straight line drawn parallel to the y-axis cuts the x-axis in T. AOF = 0 y Cc D 2 AQ 34) E B@ ; 2) a o x O T i 3.1 Calculate: 3.1.1 The length of AB. Leave your answer in surd form (2) 3.1.2 The gradient of AB (2) 3.2 Prove that OA 1 AB (2) 3.3 Determine the equation of DE. (4) 3.4 Determine the coordinates of C such that AOBC, in this order, is a-parallelogram. (3) 3.5 Calculate the: 3.5.1 Size of 0 (5) 3.5.2 Area of A ABT, if A and I are joined to form A ABT (4) [22] Copyright reserved Please turn over

mattDewatoaded from Stanmorephysics.com Grade 12 Prep. Exams. QUESTION 4 In the diagram, a circle centred at M(a ; 5) with a radius of 5 units touches the x-axis and the y-axis at points N and L, respectively. QPT is a tangent to this circle at P(-I ; 8). The coordinates.ofY are (2 ; y) 41 42 43 44 Copyright reserved T23y) Give a reason why ML | y—axis. Determine the: 4.2.1 Coordinates of M 4.2.2 Equation of the circle having centre M 4.2.3 Equation of the tangent QPT in the form y = mx +c Another circle, having point T as the centre, touches the circle, having M as the centre externally. Determine the equation of the circle centred at T in the form (x — Ay +(y—ky =r. The circle with centre M is translated across the Cartesian plane injsuch a way that both horizontal and vertical axes remain tangents to the circle/simultaneously. Write down all the possible coordinates of the centres of the newly translated circles, given that iy must be real for ALL values of x and y. FS/September 2023 (1) (2) (2) (5) (6) (4) [20] Please turn over

mattPawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. QUESTION 5 In the diagram below, the graphs of f (x) = 2cos x and g(x) = tan bx are drawn for the interval x € [90° ; 180°]. y Use the graphs to answer the following questions. 5.1 Write down the value of b. qd) 5.2 Write down the range of g for the interval x € [-90° ; 180°] . (2) 53 Write down the period of g. (1) 5.4 Write down a value of x, in the given interval, where g(x + 5°) —f(x + 5°) = 1. dd) 5.5 Write down TWO values of x in the given interval, where so 48 andefined. (2) x D 5.6 Write down the value of p if > 2cosx =0 () x= 0? [9] Copyright reserved Please turn over

mattDewatoaded from Stanmorephysics.com Grade 12 Prep. Exams. QUESTION 6 6.1 6.2 6.3 6.4 6.5 6.6 WITHOUT using a calculator, determine the following in terms of sin 25°: 6.1.1 sin.835° 6.1.2 cas‘S0P Simplify the following expression to ONE trigonometric ratio: sin(—2x).(1 — sin? x) sin(90° + x).tan x WITHOUT using a calculator, simplify (p tan30° + qsin 60°) to a single fraction in terms of p and q. Given: cos(A - B) =cos A.cosB + sin A.sinB 6.4.1 Use the formula for cos(A — B) to derive a formula for sin(A — B). 6.4.2 Prove that sin9A + sinA = 2sin5A.cos4A 6.4.3 Write down the maximum value of 37°"°4°°°** Determine the general solution of cos 2x — Scosx —2 = 0 Given: tanx = sin + yinx + /sinx + Jsinx +..., with x [oc 3 90°) WITHOUT using a calculator, show that: 2sin’ x = sin2x(cosx + 1) Copyright reserved FS/September 2023 (D) (2) (6) (3) (4) (3) (2) (6) (5) [32] Please turn over

mattPasentoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. QUESTION 7 In the diagram below, O is the centre of the circle. A, B, C and D are points on the semi- circle such that ABCD is a rectangle. The radius of the semi-circle is 6 units, COD = @ and Ol facar py (2) (3) 71 Write BOCin terms of 6 7.2 If 8 = 43°, calculate the length of BC. 73 Points A, B, C and D are shifted along the semi-circle. Calculate the value of 0 if ABCD now forms a square. (4) 9] Copyright reserved Please turn over

mattPasntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. Give reasons for your statements in QUESTIONS 8, 9 and 10. QUESTION 8 8.1 In the diagram, ABCD is a cyclic quadrilateral, and the circle has a centre O. A. Prove the theorem which states that A + C = 180°. (5) 8.2 In the diagram below, ABCD is a cyclic quadrilateral. Chords AD and BC are produced to F and E, respectively. F and E are joined such that EF || AB. Prove that CEFD is a cyclic quadrilateral. (5) [10] Copyright reserved Please turn over

mattPawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. QUESTION 9 In the diagram below, O is the centre of the circle, with points A, B and C on the circle. DCE is a tangent to the circle at C. GOC, BOJ, and GJF are straight lines. F and H are points;9n AC such that GF || OH. C, = y, O, = Aad) FH : HC = 2 : 3 9.1 Calculate, giving reasons, j, in terms of x and y. (6) 9.2 Determine, giving reasons, the value of =. (3) 19] Copyright reserved Please turn over

mattDewtoaded from Stanmorephysics.com Grade 12 Prep. Exams. QUESTION 10 In the diagram below, A, B, C, and D lie on the circle with centre O. AOFC and DFB are straight lines, DF = FB and D = x. D 10.1 Determine, with reasons, the size of EACH of the following in terms of x: 10.1.1 A 10.1.2 ¢, 10.2. Prove, giving reasons, that: 10.2.1 f =f 10.2.2 ACFB ||| ACBA 10.2.3. DC? = FC.AC 10.2.4 Copyright reserved FC_(,___ AB 1, AB AC AO +OC AO +0OC TOTAL: FS/September 2023 (2) (3) (2) (3) (4) (5) [19] 150

mattawntoaded from Stanmorephysics.com FS/September 2023 Grade 12 Prep. Exams. INFORMATION SHEET: A=P(l-ni) A=P(1-i)' — 4=P(1+i)" ;r#l s,-—4 ;-lI<r<l y=mt+e y-y=m(x--x,) m = 22 m= tan0 XH (x-a)y +(y-by =r In adBC: 2-=—2_ = © snA sinB sinC a =b’ +c —2becos A 1 : area AABC = 7 ahsin€ sin(@ + B) = sina.cos 8 + cosa.sin B sin(a — 8) =sina.cos $ — cosa.sin B cos(@ + f) = cosa.cos 8 — sina.sin B cos(a@ — f) = cosa.cos f + sina.sin B cos’ a — sin’ @ cos2@ =41—-2sin’a sin2a@ = 2sina@.cos a 2cos’ a -1 _ x y= n A P(A) = nA) n(S) p=a+bx Copyright reserved

Downloaded from Stanmorephysics.com education Department of Education FREE STATE PROVINCE NDE EKSAMEN ~“GRADE/GRAAD 12 MATHEMATICS P2/ WISKUNDE V2 SEPTEMBER 2023 MARKS/PUNTE: 150 MARKING GUIDELINES/ NASIENRIGLYNE These marking guidelines consist of 20 pages and a 3-page cognitive grid/ Hierdie nasienriglyne bestaan uit 20 bladsye en 'n 3 bladsye kognitiewe tabel. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewarboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. NOTE Marking Guidelines/Nasienriglyne FS/V'S/September 2023 If a candidate answers a question TWICE, only mark the FIRST attempt. If a candidate has crossed out an attempt of a question and not redone the question, mark the crossed-out version. Consistent-accuracy applies in ALL aspects of the marking memorandum. Stop marking the second:¢alculation error. Assumingpanswers/values in order to solve a problem is NOT acceptable. NOTA: As 'n kandidaat 'n vraag TWEE KEER beantwoord, merk slegs die EERSTE poging. As 'n kandidaat 'n antwoord van 'n vraag doodtrek en nie oordoen nie, merk die doodgetrekte poging. Volgehoue akkuraatheid word in ALLE aspekte van die nasienriglyne toegepas. Hou op nasien by die tweede berekeningsfout. Aanvaar van antwoorde/waardes om 'n probleem op te los, word NIE toegelaat nie. QUESTION/VRAAG 1 63 79 50 74 75 66 150 86 72 74 60 _ 849 Y 849 (addition of LD) x= results/ optel van uitslae) 11 Answer only: full marks = 77,18 vY answer/antwoord (CA if/as +11) (2) = v ? 1.1.2 | o = 24,86 No penalty for rounding: answer/antwoord aan 113] (¥-o:¥+0) n ofall mark ved _ ; nswer only: full marks provide Y 52,32 = (52,32 ;102,04) | 11.1 @ 1.1.2 both correct v 102,04 .. 2 results/uitslae ¥ answer/antwoord Q) 1.2 | 150 ¥ answer/antwoord Q) 13° /D ¥ Vanswer/antwoord (2) 1.4 | By doubling any of the six lowest results, the new result will be more than 74 and lie to the right hand side of 74/Deur om enige uitslag van die ses laagste uitslae te verdubbel sal die nuwe uitslag meer as 74 wees en regs van 74 lé. Because 74 is also the mode, the new median will still be | ¥ 744iode/modus 74/Omdat 74 die modus is sal die nuwe mediaan 74 bly. The median therefore has not changed/Die mediaan het dus | “no change/geen nie verander nie. verandering (2) [11] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 Temperature at midday (in °C)/ Middag- temperatuur (in--C). Number of ice 16 | 20 | 23 | 29 | 33 | 38 | 40 | 38 | 35 | 30 creams/Aantal 12 17 | 19 | 44 | 64 | 70 | 74 | 66 | 60 | 40 roomyse 2.1 | Strong/Goed Y strong/sterk The majority of the points lie close to the regression line/ v reason/rede Die meerderheid punte lé naby die regressielyn (2) OR/OF Strong/Sterk Y strong/sterk r=0,98 ¥ r= 0,98 (2) 2.2 a = ~ 38,4828... Answer only: full marks, but if a y a b = 2,8173... and b are swopped only 1/3 marks/ b Y = —38,48+2,82x | maar as a en b omgeruil is, slegs v equation/ 1/3 punte. vergelyking (3) 2.3 | » =—38,48+2,82(26) ; ¥ substitute 26 into eq./ 34.84 Answer only: full marks vervang 26 in vgl. «. 34 ice creams / roomyse Can also accept 35 v34 (2) OR/OF 26y = 34,77 (calculator / sakrekenaar) . 34 ice cream/ roomyse vv 34 (2) 2.4 | Regression line will be pulled slightly upwards/regressielyn sal | “explanation/ effe opwaarts getrek word verduideliking The prediction will be that more ice cream will be sold/ | Vmore ice cream/meer Die voorspelling is dat meer roomyse verkoop sal word. roomyse (2) 19] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 Ay e0nn 400) lon - A(2 54) 4 x (e) 3.1.1 | AB= (6-2) +(2-4) Y subst./vervang = 25 ¥ answer/antwoord (2) 3.1.2 | my = 2-4 ¥ subst./vervang 6-2 Answer only: full marks = I v answer/antwoord 2 (2) 3.2 Moy = 2 ¥ my, =2 But / maar : Moy * Map - (-3) 2 1 =-| v (-5] =-l QO Jona? (2) Tanay a Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattaswntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne 3.3 DE || AB [midpt.theorem / midpt.—stelling | YS OR/OFR 1 Mor = — 5 YMyp tvouetewr( 6:5 x-6) Y subst./vervang (6:3) + 1s ¥ equation/ 2 vergelvking (4) 34 If/ As AOBC is parm: Answer only: full marks OA || BC and/en OB || AC Xy > X_ = Xo + 6 (translation / translasie) Wy DX = 246 ¥ method/metode WX =8 Vx. =8 in the same way / op dieselfde wyse : Vy PIV =442 “Ve =6 VY Vo =6 (3) OR/OF midpt. AB = (4 ; 3) ..midpt. OA =(4;3) — [diag./ hoekln. parm.] OFX _ 4 and/en OF Ve _ 3 v method/metode 2 xX, =8 “Ve =6 Vx. =8 “C(8; 6) Vy =6 (3) OR/OF eq./vgl. BC: y = 2x — 10 1 10 eq./vgl. AC: y = =x + — q./ VS! y 3 3 1 10 Tay 2x -10==x+ 3 v method/metode 5x = 40 : x=8 and/en: y = 2(8) -10=6 “.C(8 3 6) (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne FS/V'S/September 2023 3.5.1 tan AOT = 2 ¥ tan AOT = mg, . AOT = 63,43° v answer/antwoord Y tanFOT = 3 4 ¥ answer/antwoord 0 = 63,43°- 36,87° “. @ = 26,56° ¥ answer/antwoord (5) 3.5.2 ay 7 OE) es a 3,43 oO BT = 2 units/eenhede v BT=2 L,=4 vi, =4 area AABT = $(2)(4) v¥ subst./vervanging = 4 units’ / eenhede* ¥ answer/antwoord (4) OR/OF BT =2 units/eenhede v BT=2 ABT = 116,57° [2s of quad./ Ze van vierhk.] v ABT area AABT = 5(2N8)(2)sin1 16,57° ¥ spbgtvervanging = 3,9998... = 4 units’ / eenhede* v answer/antwoord (4) OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattbawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne O Area AABT = Area AOTQ— Area A AOT—AreaA ABQ | ¥ method/metode 1 1 ] 1 = 5(4 + 6)x4 ~ 5(6)(4) - 5 (2)(4) ¥5(4 + 6)x4 = 4 units’ / eenhede’ 1 1 #4 (ay(ajaet(ay(a) ¥ answer/antwoord (4) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 4 4y T(2 5) N O a 4.1 radius | tangent / radius | raaklyn ¥ rede/reason QQ) 4.2.1 | M(-5; 5) vx =-5 Vy=5 (2) 4.2.2) (x 1 sy 1 (y sy = 25 VY (x4 sy t(y sy vr? =25 (2) 8-5 4.2.3 | My = 1-5) ¥ subst./vervanging 3 3 = 4 Y Mp = 4 : 4 v 4 “Mor 3 [rad L tangent /rad 1 raaklyn] Mex =3 8=-F(-1) +e OR/OF y=8=-3(x-(-1) ¥ subst./vervang m & (—1 ; 8) 20 c= 3 _ nn ys _4 x+ 20 v equation/vergelyking 3 3 Lonny (5) a? Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. FS/V'S/September 2023 Marking Guidelines/Nasienriglyne 43 4 20 ye 302) + 5 ¥ subst./vervanging yo4 vy, =4 T(2;4) MT = (27 (-3)/ +(4- 5)" ¥ subst./vervanging = 52 ¥ answer/antwoord radius circleT/ radius sirkelT = 5/2 — 5 v MT—5 2 2 (x - 2) + (y ~ 4) = 75 ~ 50V2 ¥ equation/vergelyking ~ 4,29 (6) 4.4 | If the x-axis and y-axis have to remain simultancously as tangents to the circle M, then the circle is reflected across the axes/As die x-as en y-as gelyktydig as raaklyne aan die sirkel M bly, dan word die sirkel oor die asse gereflekteer. The only quadrants where xy 2 0, are quadrants | and 3/ Die enigste kwadrante waar xy = 0, is kwadrante 1 en 3 M(5;5) & M“(-5;-5) | Accuracy /Akkuraatheid vv M(5;5) But CA from 4.2.1 / vv M"(-5 ;-5) Maar CA van 4.2.1 ay oM(5;5) } eo TO —— ¥ \ M'-5 ;-5) a ) [20] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

matPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. QUESTION/VRAAG 5 Marking Guidelines/Nasienriglyne x a 5.1] o= 1 ¥ answer/antwoord 2 (1) 5.2] y>-1 OR/OF ye [-1; 2) ¥ critical values/kritiese waardes Y notation/notasie (2) 5.3 | 360° v 360° () 5.4 | x = 85° ¥ 85° d) 5.5 | x= 0° or/of x= 180° v 0° v 180° (2) 5.6 | p= 180° “7 iat Ang] 2) 2200] [9] Copyright reserved/Kopiereg voorbehou Ei Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 6.1.1 | sin335° =-sin25° v¥—sin 25° trad ) 6.1.2 | cos50°lina? = cos2{2 ) Answer only: full marks ¥ cos 2(25°) =] -2sint 25 v1 — 2sin? 25 (2) 6.2 sin(—2x).(I — sin? x) sin(90° + x). lan x _ (-sin 2x)(cos* x) Y —sin2x (cosx) sinx ¥ cos’ x cos. x ¥ cosx —2sin x.cos x.cos” x vy Snax = cos x ; sin v —2sin.x.cosx =-2cos’x Y¥ —2cos? x (6) 6.3 (p tan 30° + qsin 60°)” 2 ate p= ! ] ¥ substitution/vervanging Bo -| Poy vi 3° 2 . =P 4 pgs 3q° Y expansion/witbreiding 3 4 - 4p’ + 12pq + 99° ¥ answer/antwoord 12 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne FS/VS/September 2023 6.4.1 | sin(A — B) = cos| 90° -(A- B)] - cos| (90° + B) - A] OR/OF sin(A — B) = cos| 90° - (A - B) | = cos(90%=> B).cos A + sin(90° + B).sin A = (-sinB)dos A + cosB.sinA = sin A-cosB — sinB.cosA = cos/ (90° — A) - (-B)] = cos(90° — A).cos(—B) + sin(90° — A).sin(-B) = sin A.cosB + cos A.(—sinB) = sin A.cos B — sin B.cos A ¥ co-ratio/ko-verhouding Y writing as difference of A and B/skryf as die verskil van AenB ¥ expansion/uitbreiding ¥ all reductions/alle reduksies (4) Y¥ co-ratio/ko-verhouding Y writing as difference of A and B/skryf as die verskil van AenB ¥ expansion/uitbreiding Y¥ all reductions/alle reduksies (4) 6.4.2 | LHS/LK= sin9A + sinA = sin(5A +4A) + sin(5A — 4A) = sin5A.cos4A + sin4A.cos5A + sinSA.cos4A — sin4A.cos5A| ¥ expansion/uitbreiding v sin(SA +4A) v sin(SA — 4A) = 2sin 5A.cos 4A (3) 6.4.3. | Max value of/Maks waarde van: 2 sin SA.cos 4A = 2 ¥ max value/maks. waarde .B=9 2 sin 5A.cos 4A — 2 «max value / maks. waarde ; 3°°"°*°""* = 9 v9 (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. FS/VS/September 2023 Marking Guidelines/Nasienriglyne 6.5 cos2x — 5cosx —2 =0 2cos* x —1-Scosx -2=0 ¥ 2c0s* x -1 2cos* x —Scosx -3 =0 (2cosx BH) (cosx -3)=0 ¥ factors/faktore 7 ~or /of cosx =3 ¥ both equations/heide vergls. cosx #3 VY cosx #3 x = 120° + k.360; or/of x = 240° +k.360° ke Z ¥x=120° & x = 240° ¥+k.360° ,keZ (6) 6.6 tan = yin + oinx + sinx + fsinx +... tan? x = sinx + fin + yin x + ./sinx + Jsinx +... tan’ x = sinx + tanx sin’x sin x z =smx+ cos x cos x sin’ x = sinx.cos’ x + sinx.cos x 2sin? x = 2sin x.cos’ x + 2sinx.cos.x 2sin’ x = sin2x.cos.x + sin 2x 2sin* x = sin2x(cosx + 1) v sinx + sinx +sinx +-ysinx... ¥ tan’ x = sinx + tanx sin’x sinx ¥ — =sinx + COS’ xX COSX ¥ multiply by/vermenigvuldig met 2cos? x v double angle identity/ dubbelhoekidentiteit (5) [32 Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 a B c d0nn Fi jaca 6 | A / D 7.1 | BOA =0@ VY BOA =0 BOC = 180° — 20 ¥ answer/antwoord (2) 7.2 | ROC = 94° ¥ BOC BC? = OB? + OC? — 20B.0C.cos BOC BC? = 6 + 6° — 2(6)(6)cos94° BC? = 77,0224... ¥ Subst. in cosine rule correctly/vervang korrek in cos-reel Copyright reserved/Kopiereg voorbehou «. BC = 8,78 units / eenhede Y answer/antwoord (3) 7.3 | For ABCD to be a square/Vir ABCD om vierkant te wees: AD=DC «. DC = 20D ¥ DC =20D In AODC tang = BC ¥ tang = BC OD OD _ 20D OD =2 ¥ tan@ =2 -. 6 = tan’! (2) 6 = 63,43° YO = 63,43° rant (4) Ss Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne GEOMETRY/MEETKUNDE Please read carefully through the following table before marking QUESTION 8-10/ Lees asseblief sorgvuldig deur die volgende tabel alvorens VRAAG 8-10 nagesien word. The6rder in which the candidate answers a geometry question must follow logically/Die volgorde waarin 'n kandidaat 'n meetkundevraag beantwoord moet logies-volg. Example/Voorbeeld Given/Gegee AB || CD and/en EFD = 115° A D 65 115 B Cc The candidate first needs to calculate x BEFORE he/she can calculate y/Die kandidaat moet eerste vir x bereken VOORDAT hy/sy vir y kan bereken. A mark for a correct statement (A statement mark is independent of a reason) 'n Punt vir 'n korrekte bewering (‘n Punt vir 'n_bewering is onafhanklik van die rede) A mark for the correct reason (A reason mark may only be awarded if the statement is correct) 'n Punt vir 'n_ korrekte rede (‘n Punt word slegs vir die rede toegeken as die bewering korrek is) S/R Award a mark if the statement AND reason are both correct (Both MUST be correct to get one mark) Ken 'n punt toe as die bewering EN rede beide korrek is (Beide MOET korrek wees om een punt te kry) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 8.1 Construction/Konstr.: Draw BO and OD/Trek BO en OD. Y constr./ konstr. O, = 2A [Zat centre = 2x Z at circ./midptsZ = 2x omtreks 2] ¥ S/R 6, =2C [Zat centre = 2x Z at cire./midpisZ = 2x omireks Z| vs é, + 0, = 360° [4s around a point/Ze om'n punt] VSR “2A + 2C = 360° The last statement mark / Die laaste “ 2(A + €) = 360° punt vir bewering: vs A+ =180° 2A + 2C = 360 OR/OF 2(A + C) — 360° (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmorephysics.com Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne 8.2 FS/VS/September 2023 Let/Laat: A = x 8.2 Cc, =A=x [ext.Z cyclic quad./buite Z kvh. | VSYR F =180° - A =180° - x [co-interior £s/ko — binneZe; AB||FE ] YS/IR C, + F=x + 180° — x =180° vs .«. CEFD is a cyclic quad/is 'n kvh. [converse opp.Zs cyclic quad/omgekeerde teenoorst.Ze kvh] / YR [ext.z of quad = opp. interior Z/buiteZ van vierhoek = teenoorst. binneZ] (5) OR/ OF C, =180° - A=180°-x — [opp.Zs cyclic quad/teenoorst.Ze kvh | YSYR F =180° — A =180° — x [co-interior s/ko — binneZe; AB||FE ] vS/R C, = F=180°- x YS «. CEFD is a cyclic quad/is 'n kvh. [converse ext.Z cyclic quad/omgekeerde buite.Z kvh]/ _ YR [opp.Zs of quad supplementary/teenoorst.Ze van vierhoek supplgpiener| (5) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPewntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne A QUESTION/VRAAG 9 E D 91 |¢ =A=y [tan-chord theorem/raaklyn - koordstelling ] vSVYR 0, =2A= 2y [Zat centre = 2x Z at cire./ midptsZ = 2x omtreks £] v¥SYR 0, =180°-2y-—x [Zson straight line/ Ze op reguit yn] YS/IR i, =180°-2y-—x [corresponding Zs/ooreenkomstige Ze; GF || OH] VSR (6) OR/OF C,=A-=y [tan-chord theorem/raaklyn-koordstelling | vSVYR O, = 2A = 2y [Zat centre = 2x Z at cire./ midptsZ = 2x omtreksZ] ¥SYR Ge=x [corresponding Zs / ooreenkomstige Ze; GF || OH] vS/R 0, =2y [opp. Zs =/regoorst.Ze =] J, =180°-2y - x [sum of Zs in A/som van Ze in A] vSIR — (6) 9.2 | FH_ GO [prop.th / eweredigh.st ; GF || OH] enn YSYR FC GC df =2 v answer/ 5 antwoord (3) 19] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattPawntoaded rfrom Stanmorephysics.com FS/VS/September 2023 Grade/Graad 12 Prep. Exams/Voorb. Eksam. Marking Guidelines/Nasienriglyne QUESTION/VRAAG 10 A B 1 2 (@) y 1 ‘A F 3 2 X D 10.1.1 )A=x [Zs in same segm.//e in dies. segm.] YSYR Q) 10.1.2. | ABC =90° [Zin semicircle./Z in halfsirkel] YSVYR CG = 90° —x [sum of Zs of A/som van Ze van A] VSR (3) 10.2.1 | OF | DB [line from centre to midpt.chord/midpt. sirkel, midpt.koord ] YSYR “5 =5 =90° 2) 10.2.2 | In ACFB and/en ACBA on = ¢, [common/gemeenskaplik | YS/R F, = B=90° [from 10.1.2 & 10.2.1] vS A CFB ||| ACBA [2 2 2] YR (3) OR/OF In ACFB and/ en ACBA c, =¢, [common/gemeenskaplik | YSIR f, = B= 90° [from 10.1.2 & 10.2.1] Ys B, =A [sum of Zs of A/ som van Ze van A] YS/R .. ACEB ||| ACBA (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

mattDewrboaded vfrom Stanmoreph SC/NSS — Marking ics.com uidelines/Nasienriglyne NW/September 2023 10.2.3 CF _ CB — [from / vanuit A] vS/R CB CA -. CB? = = AC.FC YS bu fit har: ADFC = AFBC [S/S] YS/R ul + De BC vs “(BE - AC.FC (4) 10.2.4 AC’ = AB’ + BC’ [Pythagoras] vS AC’ = AB’ + FC.AC [BC’ = DC’ ] ¥ BC? = FC.AC 7 AB’ FC ; AC: AC V+ AC FC) AB yFC _,_ AB’ AC AC? “ac Ac FC AB AB AC (! _ “\ + *) ¥ factorise/faktoriseer but / maar AC = AO + OC .FC_(,;__ AB \/,,_ AB “VAC AO + OC AO + OC (3) [19] TOTAL/TOTAAL: 150 40.00] Lonny re Copyright reserved/Kopiereg voorbehou

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